CS224W: Social and Information Network Analysis Jure...

CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University

http://cs224w.stanford.edu

Spreading through networks: Cascading behavior Diffusion of innovations Network effects Epidemics

Behaviors that cascade from node to node like an epidemic

Examples: Biological: Diseases via contagion

Technological: Cascading failures Spread of information

Social: Rumors, news, new

technology Viral marketing

10/18/2011 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 2


Product adoption: Senders and followers of recommendations



Behavior/contagion spreads over the edges of the network

It creates a propagation tree, i.e., cascade


Cascade (propagation graph) Network

Terminology: • Stuff that spreads: Contagion • “Infection” event: Adoption, infection, activation • We have: Infected/active nodes, adoptors

Probabilistic models: Models of influence or disease spreading An infected node tries to “push”

the contagion to an uninfected node Example: You “catch” a disease with some prob.

from each active neighbor in the network Decision based models: Models of product adoption, decision making A node observes decisions of its neighbors

and makes its own decision Example: You join demonstrations if k of your friends do so too


Collective Action [Granovetter, ‘78] Model where everyone sees everyone

else’s behavior Examples: Clapping or getting up and leaving in a theater Keeping your money or not in a stock market Neighborhoods in cities changing ethnic composition Riots, protests, strikes


[Granovetter ‘78]

n people – everyone observes all actions Each person i has a threshold ti Node i will adopt the behavior iff at

least ti other people are adopters: Small ti: early adopter Large ti: late adopter

The population is described by {t1,…,tn} F(x) … fraction of people with threshold ti ≤ x


0

1

P(a

dopt

ion)

ti

Think of the step-by-step change in number of people adopting the behavior: F(x) … fraction of people with threshold ≤ x s(t) … number of participants at time t

Easy to simulate: s(0) = 0 s(1) = F(0) s(2) = F(s(1)) = F(F(0)) s(t+1) = F(s(t)) = Ft+1(0)

Fixed point: F(x)=x There could be other fixed

points but starting from 0 we never reach them

10/18/2011 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

x

y=F(x)

y=x

11

Iterating to y=F(x). Fixed point.

F(0)

y=F(

x)

What if we start the process somewhere else? We move up/down to the next fixed point How is market going to change?


x

y=F(

x) y=x

x

x


x

y=F(

x)

y=x

Robust fixed point

Fragile fixed point

Each threshold ti is drawn independently from some distribution F(x) = Pr[thresh ≤ x] Suppose: Normal with µ=n/2, variance σ Small σ: Large σ:



Bigger variance let’s you build a bridge from early adopters to mainstream

Small σ Medium σ

F(x) F(x)

No cascades! Small cascades

Fixed point is low


But if we increase the variance even more we move the higher fixed point lover

Big σ Huge σ

Big cascades!

Fixed point gets lower!

Fixed point is high!

It does not take into account: No notion of social network – more influential

users It matters who the early adopters are, not just

how many Models people’s awareness of size of participation

not just actual number of people participating Modeling thresholds Richer distributions Deriving thresholds from more basic assumptions game theoretic models


It does not take into account: Modeling perceptions of who is adopting the

behavior/ who you believe is adopting Non monotone behavior – dropping out if too

many people adopt Similarity – thresholds not based only on numbers People get “locked in” to certain choice over a

period of time

Network matters! (next slide)


Based on 2 player coordination game 2 players – each chooses technology A or B Each person can only adopt one “behavior”, A or B You gain more payoff if your friend has adopted the

same behavior as you


[Morris 2000]

Local view of the network of node v


Payoff matrix: If both v and w adopt behavior A,

they each get payoff a>0 If v and w adopt behavior B,

they reach get payoff b>0 If v and w adopt the opposite

behaviors, they each get 0

In some large network: Each node v is playing a copy of the

game with each of its neighbors Payoff: sum of node payoffs per game



Let v have d neighbors Assume fraction p of v’s neighbors adopt A Payoffv = a∙p∙d if v chooses A

= b∙(1-p)∙d if v chooses B Thus: v chooses A if: a∙p∙d > b∙(1-p)∙d

babq+

=

Threshold: v choses A if p>q

Scenario: Graph where everyone starts with B. Small set S of early adopters of A Hard wire S – they keep using A no matter

what payoffs tell them to do

Payoffs are set in such a way that nodes say: If at least 50% of my friends are red I’ll be red (this means: a = b+ε)



If more than 50% of my friends are red I’ll be red

25

},{ vuS =


u v


26

},{ vuS =



27

u v

},{ vuS =



28

u v

},{ vuS =



29

u v

},{ vuS =



30

u v

},{ vuS =

Observation: The use of A spreads monotonically

(Nodes only switch from B to A, but never back to B) Why? Proof sketch: Nodes keep switching from B to A: B→A Now, suppose some node switched back from A→B,

consider the first node v to do so (say at time t) Earlier at time t’ (t’<t) the same node v switched B→A So at time t’ v was above threshold for A But up to time t no node switched back to B, so node v

could only had more neighbors who used A at time t compared to t’. There was no reason for v to switch.


!! Contradiction !!

Consider infinite graph G (but each node has finite number of neighbors)

We say that a finite set S causes a cascade in G with threshold q if, when S adopts A, eventually every node adopts A

Example: Path


babq+

=

v choses A if p>q

If q<1/2 then cascade occurs

S


S

S


Infinite Tree:

Infinite Grid:


Def: The cascade capacity of a graph G is the largest q

for which some finite set S can cause a cascade Fact: There is no G where cascade capacity > ½

Proof idea: Suppose such G exists: q>½,

finite S causes cascade Show contradiction: Argue that

nodes stop switching after a finite # of steps


X

Fact: There is no G where cascade capacity > ½ Proof sketch: Suppose such G exists: q>½, finite S causes cascade Contradiction: Switching stops after a finite # of steps Define “potential energy” Argue that it starts finite (non-negative)

and strictly decreases at every step “Energy”: = |dout(X)|

|dout(X)| := # of outgoing edges of active set X The only nodes that switch have a

strict majority of its neighbors in S |dout(X)| strictly decreases It can do so only a finite number of steps


X

What prevents cascades from spreading? Def: Cluster of density ρ is a set of nodes C

where each node in the set has at least ρ fraction of edges in C.


ρ=3/5 ρ=2/3

Let S be an initial set of adopters of A All nodes apply threshold q to decide

whether to switch to A Two facts: 1) If G\S contains a cluster of density >(1-q)

then S can not cause a cascade 2) If S fails to create a cascade, then

there is a cluster of density >(1-q) in G\S


So far: Behaviors A and B compete Can only get utility from neighbors of same

behavior: A-A get a, B-B get b, A-B get 0 Let’s add extra strategy “A-B” AB-A: gets a AB-B: gets b AB-AB: gets max(a, b) Also: Some cost c for the effort of maintaining

both strategies (summed over all interactions)


Every node in an infinite network starts with B Then a finite set S initially adopts A Run the model for t=1,2,3,… Each node selects behavior that will optimize

payoff (given what its neighbors did in at time t-1)

How will nodes switch from B to A or AB?


B A A AB a a a+b-c AB b

Edge payoff

Path: Start with all Bs, a>b (A is better) One node switches to A – what happens? With just A, B: A spreads if b ≤ a With A, B, AB: Does A spread?

Assume a=2, b=3, c=1


B A A a=2

B B 0 b=3 b=3

B A A a=2

B B a=2 b=3 b=3

AB

-1

Cascade stops

Let a=5, b=3, c=1


B A A a=5

B B 0 b=3 b=3

B A A a=5

B B a=5 b=3 b=3

AB

-1

B A A a=5

B B a=5 a=5 b=3

AB

-1

AB

-1

A A A a=5

B B a=5 a=5 b=3

AB

-1

AB

-1

Infinite path, start with all Bs Payoffs: A:a, B:1, AB:a+1-c What does node w in A-w-B do?


a

c

1

1

B vs A

AB vs A

w A B

AB vs B

B

B AB AB

A

A

Payoffs: A:a, B:2, AB:a+2-c Notice: now also AB spreads What does node w in AB-w-B do?


w AB B

a

c

1

1

B vs A

AB vs A

AB vs B

B

B AB AB

A

A

2

Joining the two pictures:


a

c

1

1

B

AB B→AB → A

A

2

You manufacture default B and new/better A comes along: Infiltration: If you make B

too compatible then people will take on both and then drop the worse one (B) Direct conquest: If A makes

itself not compatible – people on the border must choose. They pick the better one (A) Buffer zone: If you choose an

optimal level then you keep a static “buffer” between A and B


a

c

B stays

B→AB B→AB → A

A spreads B → A

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